arXiv subjects

Gen Tanigawa

Publications and source records attributed to Gen Tanigawa.

3 recordsLinked to original sources

arXiv2026

Contravariantly finite resolving subcategories over commutative local rings are trivial ones

We show that a contravariantly finite resolving subcategory over a henselian local ring must be the category of free modules or the whole module category or the subcategory of maximal Cohen--Macaulay modules. This removes the Gorenstein assumption from a theorem of Takahashi. We also show that a resolving subcategory of finite type other than the subcategory of free modules must be the subcategory of maximal Cohen--Macaulay modules. As a consequence, every Cohen--Macaulay local ring of finite CM type is uniformly dominant, thereby establishing a stronger form of a conjecture of Takahashi.

math.AC↗

arXiv2026

Torsion pairs in subcategories of modules over a commutative ring

Let $R$ be a commutative noetherian ring. Denote by $\operatorname{mod} R$ the category of finitely generated $R$-modules. Let $\Delta$ be a subset of $\operatorname{Spec} R$, and let $\operatorname{Ass}^{-1}\Delta$ stand for the full subcategory of $\operatorname{mod} R$ consisting of finitely generated $R$-modules whose associated prime ideals belong to $\Delta$. In this paper, we consider classifying torsion pairs in $\operatorname{Ass}^{-1}\Delta$ and some other full subcategories of $\operatorname{mod} R$.

math.AC↗

arXiv2026

Contravariantly infinite resolving subcategories

Let $R$ be a commutative Noetherian ring. Denote by $\textrm{mod}R$ the category of finitely generated $R$-modules. In this paper, a contravariantly infinite subcategory of $\textrm{mod}R$ is defined as a full subcategory $\mathscr{X}$ of $\textrm{mod}R$ such that no module outside $\mathscr{X}$ admits a right $\mathscr{X}$-approximation. This paper provides several criteria for contravariant infiniteness in the case where $R$ is a local complete intersection.

math.AC↗